Arctan Formula Bucket

by Jörg Arndt

Notation: m[x] denotes m arctan \frac{1}{x}
\frac{π}{4} = +24 [12943] − 12 [682] + 7 [239] + 44 [57]  (1)

{5, 13, 61}

− \frac{π}{4} = +115 [18543] − 237 [2943] + 301 [993] − 186 [882] − 142 [829] − 198 [378] − 29 [268]  (2)

{5, 13, 17, 29, 37, 41}

− \frac{π}{4} = +144 [18543] − 324 [2943] + 388 [993] − 244 [882] − 200 [829] − 256 [378] + 29 [239]  (3)

{5, 13, 17, 29, 37, 41}

− \frac{π}{4} = +51 [390112] + 122 [12943] + 58 [5357] − 241 [3141] − 241 [798] − 32 [682] − 176 [463]  (4)

{5, 13, 17, 61, 97, 101}

− \frac{π}{4} = +51 [390112] + 122 [12943] + 58 [5357] − 32 [682] + 65 [463] − 241 [268]  (5)

{5, 13, 17, 61, 97}

− \frac{π}{4} = +32 [6826318] − 32 [6507] + 12 [3957] − 44 [905] − 68 [268] − 115 [239]  (6)

{5, 13, 17, 109, 457}

− \frac{π}{4} = +56 [173932] + 32 [1068] − 100 [515] − 39 [239] − 88 [192]  (7)

{5, 13, 73, 101}

− \frac{π}{4} = +88 [6826318] − 24 [12943] − 88 [6050] + 56 [682] − 44 [515] − 183 [239]  (8)

{5, 13, 61, 101, 457}

− \frac{π}{4} = +56 [3014557] + 44 [6118] + 276 [2943] − 144 [1303] − 200 [818] − 127 [239]  (9)

{5, 13, 41, 53, 101}

\frac{π}{4} = +44 [18543] − 12 [882] + 56 [327] + 64 [307] + 68 [268] + 39 [239]  (10)

{5, 13, 17, 29, 37}

\frac{π}{4} = +39 [44179] + 51 [34208] + 95 [1143] + 32 [447] + 115 [307] + 68 [268]  (11)

{5, 13, 17, 29, 53}

\frac{π}{4} = +4 [39307] − 2 [23382] − 4 [9043] + 4 [307] − [239] + 14 [18]  (12)

{5, 13, 29, 97, 1297}

\frac{π}{4} = +12 [4443] + 24 [2582] + 24 [1068] − 5 [239] + 44 [57]  (13)

{5, 13, 73, 281, 677}

− \frac{π}{4} = +88 [6826318] − 24 [12943] − 88 [6050] + 56 [682] − 44 [515] − 183 [239]  (14)

{5, 13, 61, 101, 457, 1201}

− \frac{π}{4} = +44 [500150] − 88 [28322] + 71 [12943] + 44 [2943] − 139 [1068] − 171 [682] − 227 [538]  (15)

{5, 13, 41, 61, 73, 137}

− \frac{π}{4} = +44 [500150] − 88 [28322] − 156 [12943] + 44 [2943] + 88 [1068] + 56 [682] − 227 [239]  (16)

{5, 13, 41, 61, 73, 137}

−4. π = +23 [500150] − 46 [28322] + [12943] + 23 [2943] + 46 [1068] − 12 [682] − 227 [18]  (17)

{5, 13, 41, 61, 73, 137}

− \frac{π}{4} = +44 [470743] − 12 [110443] − 32 [1152] − 56 [693] − 144 [437] − 83 [239]  (18)

{5, 13, 17, 113, 1201}

− \frac{π}{4} = +68 [1999509] + 24 [58911] + 12 [12238] + 4 [307] − 95 [239] − 68 [169]  (19)

{5, 13, 29, 421, 14281}

\frac{π}{4} = +44 [4662] − 32 [4443] − 20 [1568] − 64 [1458] + 132 [268] + 83 [239]  (20)

{5, 13, 17, 89, 281}

− \frac{π}{4} = +183 [1493208] + 366 [83270] − 132 [32807] + 349 [12943] + 139 [4662] − 827 [1568] − 215 [682]  (21)

{5, 13, 17, 61, 89, 233}

− \frac{π}{4} = +51 [32807] + 166 [12943] − 44 [4662] − 95 [1568] − 32 [682] − 183 [268]  (22)

{5, 13, 17, 61, 89}

\frac{π}{4} = +32 [1493208] + 64 [83270] + 132 [32807] + 296 [12943] + 76 [4662] − 248 [1568] + 215 [239]  (23)

{5, 13, 17, 61, 89, 233}

\frac{π}{4} = +132 [32807] + 200 [12943] + 44 [4662] − 88 [1568] + 32 [682] + 183 [239]  (24)

{5, 13, 17, 61, 89}

− \frac{π}{4} = +12 [12238] − 7 [239] + 4 [57] − 24 [29]  (25)

{5, 13, 421, 1597}

\frac{π}{4} = +8 [11653127] + 4 [2413658] − 11 [1393] − 26 [577] − 4 [307] + 6 [7]  (26)

{5, 13, 29, 197, 14281}

−1.25 π = +8 [11653127] + 4 [2413658] − 11 [1393] − 26 [577] − 4 [307] − 12 [3]  (27)

{5, 13, 29, 197, 14281}

−0.75 π = +4 [58911] + 36 [12238] + 12 [307] − 13 [239] − 68 [29]  (28)

{5, 13, 29, 421}

− \frac{π}{4} = +32 [534568] + 64 [6898] − 12 [4662] + 52 [1568] − 164 [268] − 51 [239]  (29)

{5, 13, 17, 89, 149}

\frac{π}{4} = +139 [44179] + 51 [34208] + 68 [12943] + 132 [2673] + 95 [1143] + 32 [682] + 183 [307]  (30)

{5, 13, 17, 29, 53, 61}

− \frac{π}{4} = +44 [44179] + 132 [34208] − 68 [12943] − 132 [2673] + 88 [1143] − 32 [682] − 183 [239]  (31)

{5, 13, 17, 29, 53, 61}

\frac{π}{4} = +17 [12943] + 88 [3039] + 88 [2673] + 44 [2309] + 51 [1068] + 215 [682] + 183 [538]  (32)

{5, 13, 17, 53, 73, 61}

\frac{π}{4} = +12 [1068] − 12 [557] + 12 [268] − 5 [239] + 44 [57]  (33)

{5, 13, 17, 73}

\frac{π}{4} = +17 [12943] + 7 [1068] − 5 [682] + 7 [538] + 44 [57]  (34)

{5, 13, 17, 61, 73}

\frac{π}{4} = +183 [534568] − 51 [32807] + 227 [24331] − 532 [12943] + 366 [6898] + 227 [5767] + 644 [1568] + 215 [682]  (35)

{5, 13, 17, 61, 89, 101, 149}

\frac{π}{4} = +183 [534568] − 51 [32807] − 532 [12943] + 366 [6898] + 227 [4662] + 644 [1568] + 215 [682]  (36)

{5, 13, 17, 61, 89, 149}

− \frac{π}{4} = +32 [534568] + 51 [32807] − 113 [12943] + 64 [6898] − 12 [4662] + 216 [1568] − 215 [233]  (37)

{5, 13, 17, 61, 89, 149}

\frac{π}{4} = +12 [110443] − 5 [239] + 32 [57] + 12 [49]  (38)

{5, 13, 89, 149, 1201}

− \frac{π}{4} = +5 [534568] − 12 [110443] + 10 [193] − 17 [57] − 12 [49] − 10 [34]  (39)

{5, 13, 89, 149, 1201}

− \frac{π}{4} = +24 [58911] + 12 [12238] + 4 [307] − 27 [239] − 68 [99]  (40)

{5, 13, 29, 269, 421}

− \frac{π}{4} = +12 [12238] − 7 [239] + 4 [187] + 4 [82] − 24 [29]  (41)

{5, 13, 269, 421}

\frac{π}{4} = +8 [307] + 3 [239] + 8 [99] + 12 [18]  (42)

{5, 13, 29}

− \frac{π}{4} = +4 [157] − 7 [41] − 2 [38] − 10 [17]  (43)

{5, 17, 29}

− \frac{π}{4} = +8 [2943] − 3 [239] − 8 [73] − 12 [18]  (44)

{5, 13, 41}

− \frac{π}{4} = +4 [515] + [239] − 8 [10]  (45)

{5, 13, 101}

\frac{π}{4} = +4 [1985] + 4 [421] + 3 [239] + 16 [21]  (46)

{13, 17, 401}

\frac{π}{4} = +2 [123] + 2 [55] + 4 [13] + 3 [7]  (47)

{5, 17, 89}

\frac{π}{4} = +12 [322] + 7 [239] + 12 [144] − 4 [57] + 24 [34]  (48)

{5, 13, 89, 233}

− \frac{π}{4} = +12 [19703] − 24 [568] − 20 [307] − 27 [239] − 56 [99]  (49)

{5, 13, 29, 89}

− \frac{π}{4} = +2 [3940598] − 4 [199] − 4 [123] − 4 [13] − 3 [7]  (50)

{5, 17, 89, 19801}

− \frac{π}{4} = +12 [39307] − 12 [4193] − 32 [307] − 27 [239] − 56 [99]  (51)

{5, 13, 29, 97}

\frac{π}{4} = +88 [39307] + 68 [12943] − 44 [5357] − 56 [682] + 88 [307] + 139 [239]  (52)

{5, 13, 29, 61, 97}

\frac{π}{4} = +132 [32807] + 44 [19703] + 112 [12943] + 76 [682] + 44 [568] + 139 [239]  (53)

{5, 13, 29, 61, 89}

\frac{π}{4} = +200 [12943] − 132 [9466] + 220 [2917] + 176 [882] + 76 [682] + 95 [239]  (54)

{5, 13, 29, 37, 61}

\frac{π}{4} = +139 [157318] + 227 [32807] − 171 [17923] + 156 [12943] + 259 [5507] + 44 [5357] + 537 [746]  (55)

{5, 13, 37, 61, 89, 97}

\frac{π}{4} = +139 [157318] + 227 [32807] + 366 [17923] + 156 [12943] − 278 [5507] + 44 [5357] + 537 [682]  (56)

{5, 13, 37, 61, 89, 97}

− \frac{π}{4} = +40 [157318] − 48 [32807] − 8 [17923] − 156 [12943] − 80 [5507] − 44 [5357] − 179 [239]  (57)

{5, 13, 37, 61, 89, 97}

\frac{π}{4} = +88 [32807] + 88 [17923] + 156 [12943] + 44 [5357] + 120 [682] + 139 [239]  (58)

{5, 13, 61, 89, 97}

− \frac{π}{4} = +51 [32807] − 88 [17923] − 17 [12943] − 44 [5357] − 120 [682] − 139 [233]  (59)

{5, 13, 61, 89, 97}

−0.75 π = +14 [32807] + 14 [17923] − 51 [12943] + 7 [5357] + 57 [682] − 139 [57]  (60)

{5, 13, 61, 89, 97}

4. π = +18 [32807] + 18 [17923] − 6 [12943] + 9 [5357] − 26 [682] + 139 [11]  (61)

{5, 13, 61, 89, 97}

1. π = +19 [157318] + 13 [32807] + 32 [17923] + 87 [12943] − 38 [5507] − 3 [5357] + 179 [57]  (62)

{5, 13, 37, 61, 89, 97}

15.5 π = +26 [157318] + 112 [32807] + 138 [17923] + 6 [12943] − 52 [5507] + 43 [5357] + 537 [11]  (63)

{5, 13, 37, 61, 89, 97}

\frac{π}{4} = +200 [12943] − 132 [9466] + 220 [2917] + 176 [882] + 76 [682] + 95 [239]  (64)

{5, 13, 29, 37, 61}

\frac{π}{4} = +183 [157318] + 315 [32807] + 200 [12943] + 44 [4662] − 88 [1568] + 366 [746] + 215 [682]  (65)

{5, 13, 17, 37, 61, 89}

\frac{π}{4} = +11 [157318] + 14 [32807] + 28 [12943] + [4662] − 2 [1568] + 22 [746] + 43 [57]  (66)

{5, 13, 17, 37, 61, 89}

\frac{π}{4} = +112 [58911] − 68 [39307] + 56 [12238] + 68 [4193] + 200 [307] + 27 [239]  (67)

{5, 13, 29, 97, 421}

− \frac{π}{4} = +32 [3323] − 68 [3141] − 12 [616] − 64 [307] − 112 [293] − 39 [239]  (68)

{5, 13, 17, 29, 101}

− \frac{π}{4} = +56 [3014557] + 44 [6118] + 276 [2943] − 144 [1303] − 200 [818] − 127 [239]  (69)

{5, 13, 41, 53, 101}

− \frac{π}{4} = +183 [3014557] + 254 [12943] + 44 [6118] + 403 [2943] − 271 [1303] − 454 [818] − 127 [682]  (70)

{5, 13, 41, 53, 61, 101}

−3.75 π = +78 [3014557] + 127 [12943] + 25 [6118] + 203 [2943] − 128 [1303] − 206 [818] − 127 [11]  (71)

{5, 13, 41, 53, 61, 101}

− \frac{π}{4} = +13 [485298] + 8 [44179] + 2 [34208] − 10 [1143] − 24 [931] − 11 [882] − 17 [23]  (72)

{5, 13, 17, 29, 37, 53}

\frac{π}{4} = +8 [307] + 3 [239] + 8 [99] + 12 [18]  (73)

{5, 13, 29, 5741, 33461}

\frac{π}{4} = +88 [39307] + 68 [12943] − 44 [5357] − 56 [682] + 88 [307] + 139 [239]  (74)

{5, 13, 29, 61, 97}

− \frac{π}{4} = +71 [12943] − 44 [5357] − 51 [2943] + 88 [2059] + 56 [682] − 227 [255]  (75)

{5, 13, 41, 61, 97}

0.5 π = +85 [12943] + 88 [5357] + 102 [2943] + 51 [2059] + 115 [682] + 227 [172]  (76)

{5, 13, 41, 61, 97}

− \frac{π}{4} = +71 [12943] + 227 [6118] + 183 [5357] − 51 [2943] − 366 [2059] + 56 [682] − 454 [606]  (77)

{5, 13, 41, 53, 61, 97}

33.5 π = +20 [12943] − 6 [5357] + 24 [2943] + 12 [2059] − 13 [682] + 227 [2]  (78)

{5, 13, 41, 53, 61, 97}

\frac{π}{4} = +398 [390112] − 449 [32807] − 32 [17923] − 779 [12943] + 183 [5357] + 657 [4662] + 1074 [1568]  (79)

{5, 13, 17, 61, 89, 97}

− \frac{π}{4} = +139 [390112] − 88 [32807] − 505 [17923] − 295 [12943] − 183 [5357] + 417 [4662] − 537 [682]  (80)

{5, 13, 17, 61, 89, 97}

−1.25 π = +158 [390112] + 97 [32807] + 160 [17923] + 673 [12943] + 159 [5357] − 63 [4662] − 1074 [268]  (81)

{5, 13, 17, 61, 89, 97}

\frac{π}{4} = +40 [390112] + 88 [32807] − 32 [17923] + 116 [12943] + 4 [5357] + 120 [4662] + 179 [239]  (82)

{5, 13, 17, 61, 89, 97}

\frac{π}{4} = +40 [390112] − 91 [32807] − 32 [17923] − 63 [12943] + 4 [5357] + 120 [4662] + 179 [233]  (83)

{5, 13, 17, 61, 89, 97}

− \frac{π}{4} = +51 [390112] + 122 [12943] + 58 [5357] − 32 [682] + 65 [463] − 241 [268]  (84)

{5, 13, 17, 61, 97}

− \frac{π}{4} = +51 [32807] + 166 [12943] − 44 [4662] − 95 [1568] − 32 [682] − 183 [268]  (85)

{5, 13, 17, 61, 89}

\frac{π}{4} = +88 [32807] + 88 [17923] + 156 [12943] + 44 [5357] + 120 [682] + 139 [239]  (86)

{5, 13, 61, 89, 97}

\frac{π}{4} = +132 [32807] − 234 [19703] + 112 [12943] − 139 [6107] + 278 [1636] + 215 [682] + 183 [568]  (87)

{5, 13, 17, 29, 61, 89}

− \frac{π}{4} = +12 [39307] − 12 [4193] − 32 [307] − 27 [239] − 56 [99]  (88)

{5, 13, 29, 97}

− \frac{π}{4} = +83 [390112] + 122 [39307] − 305 [4246] − 151 [4193] + 193 [463] − 337 [307]  (89)

{5, 13, 17, 29, 97}

− \frac{π}{4} = +12 [19703] − 24 [568] − 20 [307] − 27 [239] − 56 [99]  (90)

{5, 13, 29, 89}

\frac{π}{4} = +24 [3323] + 12 [515] + 8 [307] + 39 [239] + 56 [99]  (91)

{5, 13, 29, 101}

− \frac{π}{4} = +24 [882] − 20 [307] − 39 [239] + 12 [117] − 68 [99]  (92)

{5, 13, 29, 37}

−0.75 π = +4 [39307] + 3 [6962] + 5 [5507] + 8 [4193] + [2436] + 4 [1433] − 19 [8]  (93)

{5, 13, 29, 37, 73, 97}

\frac{π}{4} = +51 [114483] + 68 [12943] − 32 [12238] + 88 [3957] + 115 [1292] + 44 [905] + 183 [294]  (94)

{5, 13, 17, 61, 109, 421}

\frac{π}{4} = +183 [534568] − 51 [32807] − 532 [12943] + 366 [6898] + 227 [4662] + 644 [1568] + 215 [682]  (95)

{5, 13, 17, 61, 89, 149}

\frac{π}{4} = +366 [1049433] + 132 [32807] − 166 [12943] − 322 [4662] + 644 [1568] − 183 [1393] + 398 [682]  (96)

{5, 13, 17, 61, 89, 197}

− \frac{π}{4} = +68 [11653127] + 68 [1999509] + 24 [58911] + 12 [12238] − 163 [1393] − 394 [577] + 4 [307]  (97)

{5, 13, 29, 197, 421, 14281}

\frac{π}{4} = +88 [32807] + 88 [17923] + 156 [12943] + 44 [5357] + 120 [682] + 139 [239]  (98)

{5, 13, 61, 89, 97}

− \frac{π}{4} = +51 [32807] − 88 [17923] − 17 [12943] − 44 [5357] − 120 [682] − 139 [233]  (99)

{5, 13, 61, 89, 97}

−0.75 π = +14 [32807] + 14 [17923] − 51 [12943] + 7 [5357] + 57 [682] − 139 [57]  (100)

{5, 13, 61, 89, 97}

− \frac{π}{4} = +278 [1049433] + 190 [32807] − 644 [17923] − 156 [12943] − 322 [5357] − 139 [1393] − 398 [682]  (101)

{5, 13, 61, 89, 97, 197}

\frac{π}{4} = +120 [1049433] + 208 [32807] − 152 [17923] + 156 [12943] − 76 [5357] + 139 [1393] + 398 [577]  (102)

{5, 13, 61, 89, 97, 197}

\frac{π}{4} = +120 [1049433] + 9 [32807] − 152 [17923] − 43 [12943] − 76 [5357] − 60 [1393] + 199 [233]  (103)

{5, 13, 61, 89, 97, 197}

− \frac{π}{4} = +56 [173932] − 39 [2436] + 32 [1068] − 88 [697] − 100 [515] − 127 [265]  (104)

{5, 13, 37, 73, 101}

\frac{π}{4} = +24 [268] + 7 [239] + 20 [57] + 12 [38]  (105)

{5, 13, 17}

\frac{π}{4} = −5 [239] + 8 [57] + 12 [18]  (106)

{5, 13}

\frac{π}{4} = +68 [12943] + 44 [5357] − 56 [682] + 88 [560] + 51 [239] + 88 [182]  (107)

{5, 13, 53, 61, 97}

\frac{π}{4} = +398 [390112] − 449 [32807] − 32 [17923] − 779 [12943] + 183 [5357] + 657 [4662] + 1074 [1568]  (108)

{5, 13, 17, 61, 89, 97}

\frac{π}{4} = −24 [58911] − 12 [12238] − 4 [307] + 27 [239] + 68 [99]  (109)

{5, 13, 29, 421}

0.5 π = −241 [2827300] + 139 [390112] + 238 [12943] + 366 [5357] + 305 [2207] + 64 [987] + 593 [463]  (110)

{5, 13, 17, 61, 97, 1597}

\frac{π}{4} = +68 [2943] + 68 [993] + 44 [882] + 88 [307] − 29 [239] + 56 [117]  (111)

{5, 13, 29, 37, 41}

\frac{π}{4} = −24 [882] + 20 [307] + 39 [239] − 12 [117] + 68 [99]  (112)

{5, 13, 29, 37}

\frac{π}{4} = −20 [50770] + 15 [239] + 20 [207] + 32 [117] + 24 [68]  (113)

{5, 13, 37, 857}

\frac{π}{4} = −6 [339877] + 12 [1755] + 18 [443] + 18 [268] + [239] + 38 [57]  (114)

{5, 13, 17, 157, 577}

\frac{π}{4} = −7 [8827] + 3 [2682] + 5 [38] + 17 [26]  (115)

{5, 17, 677}

\frac{π}{4} = +3 [11420868] − 3 [15568] − 3 [293] − 6 [58] + 5 [38] + 10 [13]  (116)

{5, 17, 101, 673, 6709}

\frac{π}{4} = −12 [93707] + 24 [12943] − 12 [687] + 7 [239] + 44 [57]  (117)

{5, 13, 61, 109, 433}

\frac{π}{4} = −8 [4030] − 4 [515] − [239] + 16 [20]  (118)

{5, 13, 101, 401}

\frac{π}{4} = −24 [119257] + 44 [14942] + 44 [2621] − 48 [707] − 5 [239] + 100 [117]  (119)

{5, 13, 37, 193, 769}

0.5 π = −34 [27493] − 17 [3458] − 3 [1068] − 14 [557] + 61 [38]  (120)

{5, 17, 73, 181}

\frac{π}{4} = +44 [27493] + 83 [3458] − 32 [1068] + 115 [557] + 122 [343] + 61 [268]  (121)

{5, 13, 17, 73, 181}

\frac{π}{4} = −8 [16480443] − 16 [11653127] − 8 [2827807] + 19 [1393] + 46 [577] + 8 [307] + 12 [18]  (122)

{5, 13, 29, 197, 457, 14281}

\frac{π}{4} = −6 [390112] + 6 [463] + 42 [268] + 13 [239] + 32 [57]  (123)

{5, 13, 17, 97}

\frac{π}{4} = +266 [23747457] − 561 [2733307] + 632 [390112] + 593 [6962] − 417 [5507] − 1635 [5283] + 1308 [2436] + 808 [1477]  (124)

{5, 13, 17, 37, 73, 97, 173}

\frac{π}{4} = −12 [83387] + 44 [58911] + 56 [14343] + 64 [307] + 27 [239] + 68 [148]  (125)

{5, 13, 29, 421, 337, 641}

\frac{π}{4} = +200 [12943] − 132 [9466] + 220 [2917] + 95 [1393] + 176 [882] + 76 [682] + 190 [577]  (126)

{5, 13, 29, 37, 61, 197}

\frac{π}{4} = +10 [12943] + 7 [4443] + 14 [2582] + 14 [1068] − 5 [682] + 44 [57]  (127)

{5, 13, 61, 73, 281}

\frac{π}{4} = +12 [515] + 7 [239] − 24 [212] + 8 [57] + 24 [34]  (128)

{5, 13, 89, 101}

\frac{π}{4} = +68 [12943] + 44 [5357] + 32 [682] + 51 [239] + 88 [172]  (129)

{5, 13, 61, 97}

\frac{π}{4} = −10 [11018] − 5 [1393] + 2 [443] + 22 [28]  (130)

{5, 157, 197}

\frac{π}{4} = +12 [11018] − 5 [1393] − 22 [577] + 24 [443] + 44 [57]  (131)

{5, 13, 157, 197}

\frac{π}{4} = −10 [11018] + 17 [1393] − 20 [443] + 44 [183] + 22 [38]  (132)

{5, 17, 157, 197}

\frac{π}{4} = −10 [11018] − 5 [1393] + 2 [443] + 22 [28]  (133)

{5, 53, 157, 197}

\frac{π}{4} = +34 [11018] + 22 [6072] + 44 [5118] + 17 [1393] + 68 [443] + 110 [182]  (134)

{5, 53, 157, 197, 373}

\frac{π}{4} = −10 [10133] − 4 [157] + 17 [41] + 2 [38] + 10 [29]  (135)

{5, 17, 29, 421}

\frac{π}{4} = −10 [13443] − 10 [10133] − 10 [293] − 24 [157] + 27 [41] + 12 [38]  (136)

{5, 17, 29, 101, 421}

\frac{π}{4} = +112 [58911] + 56 [12238] + 136 [3323] + 68 [515] + 64 [307] + 95 [239]  (137)

{5, 13, 29, 101, 421}

\frac{π}{4} = −11 [10133] + [818] + [394] − 2 [182] + 17 [41] + 11 [29]  (138)

{5, 53, 29, 101, 421}

\frac{π}{4} = +88 [32807] + 88 [17923] + 156 [12943] + 44 [5357] + 120 [682] + 139 [239]  (139)

{5, 13, 61, 89, 97}

\frac{π}{4} = +71 [58911] + 42 [12238] + 122 [2943] − 13 [1713] + 27 [829] + 41 [378] + 186 [307]  (140)

{5, 13, 17, 29, 41, 421}

\frac{π}{4} = −44 [17557] − 115 [12943] − 359 [5257] + 7 [2943] − 44 [829] + 215 [682] + 227 [378]  (141)

{5, 13, 17, 29, 41, 61}

\frac{π}{4} = +20 [307] + 44 [268] + 27 [239] + 20 [157] + 12 [38]  (142)

{5, 13, 17, 29}

\frac{π}{4} = +12 [443] − 5 [239] + 20 [57] + 12 [28]  (143)

{5, 13, 157}

\frac{π}{4} = −10 [11018] − 5 [1393] + 2 [1068] + 2 [757] + 22 [28]  (144)

{5, 73, 157, 197}

\frac{π}{4} = −24 [5357] − 36 [682] − 17 [239] − 48 [172] + 68 [57]  (145)

{5, 13, 61, 97}

\frac{π}{4} = +24 [12943] − 12 [682] + 7 [239] + 44 [57]  (146)

{5, 13, 61, 97}

\frac{π}{4} = −[239] + 4 [5]  (147)

{13, 601, 877}

\frac{π}{4} = −4 [1710] − 4 [580] − [239] + 12 [15]  (148)

{13, 113, 229}

\frac{π}{4} = +8 [307] + 3 [239] + 8 [99] + 12 [18]  (149)

{5, 13, 29}

\frac{π}{4} = −12 [21637] + 48 [1113] + 36 [353] + 19 [239] + 32 [57]  (150)

{5, 13, 17, 733}

\frac{π}{4} = −12 [358307] + 39 [4633] + 32 [307] + 27 [252] + 56 [99]  (151)

{5, 13, 29, 977}

\frac{π}{4} = +24 [12943] − 12 [682] + 7 [239] + 44 [57]  (152)

{5, 13, 61, 349}

\frac{π}{4} = −12 [19287] + 24 [12943] + 32 [707] + 7 [239] + 44 [62]  (153)

{5, 13, 61, 769}

\frac{π}{4} = −12 [3112] + 24 [1068] + 12 [757] − 5 [239] + 44 [57]  (154)

{5, 13, 73, 157}

\frac{π}{4} = +7 [1068] + 27 [557] + 17 [378] + 34 [319] + 22 [38]  (155)

{5, 17, 41, 73}

\frac{π}{4} = +3 [17923] − 7 [5357] + [682] + 3 [233] + 17 [22]  (156)

{5, 61, 89, 97}

\frac{π}{4} = −7 [5357] − 2 [682] + 3 [172] + 17 [22]  (157)

{5, 61, 97, 349}

\frac{π}{4} = +3 [17923] − 7 [5357] + [682] + 3 [233] + 17 [22]  (158)

{5, 61, 89, 97, 349}

\frac{π}{4} = −7 [5357] − 2 [4403] − 2 [807] + 3 [172] + 17 [22]  (159)

{5, 61, 97, 521}

\frac{π}{4} = −10 [11018] − 5 [1393] + 2 [1068] + 2 [757] + 22 [28]  (160)

{5, 73, 157, 197}

\frac{π}{4} = −10 [11018] − 5 [1393] + 2 [443] + 22 [28]  (161)

{5, 61, 157, 197}

\frac{π}{4} = +2 [12218] + 2 [1068] + 2 [807] + 2 [28] + 5 [7]  (162)

{5, 73, 157, 521}

\frac{π}{4} = −22 [87892] − 10 [11018] − 5 [1393] + 2 [443] + 44 [56]  (163)

{5, 157, 197, 3137}

\frac{π}{4} = −5 [8119] − 10 [3363] − 10 [577] + 12 [117] + 24 [68] + 20 [57]  (164)

{5, 13, 37, 197, 5741, 34261}

\frac{π}{4} = +17 [12943] − 12 [8307] − 5 [743] + 7 [343] + 44 [57]  (165)

{5, 13, 61, 181}